# Finitary mono preserving functors on varieties that fail to preserve intersections.

Let $Alg$ be a single-sorted finitary variety i.e. one specified by a set $\Sigma$ of operations with finite arity and also a set $E$ of equations, the morphisms being the usual $\Sigma$-algebra morphisms.

Further let $T : Alg \to Alg$ be a finitary functor that preserves monos (injections) and empty binary intersections i.e. $T(A \cap B) = TA \cap TB$ whenever $A \cap B = \emptyset$. This trivially holds if $\Sigma$ contains a constant.

Then my question is:

Does there exist such a $T$ that fails to preserve finite or infinite intersections?

Note that every $T : Set \to Set$ preserves non-empty finite intersections by page 3 of:

so under my assumptions $T$ would preserve finite intersections. Then if $T$ is finitary this can be shown to imply preservation of infinite intersections. Therefore no such example can be found in $Set$, this being the variety where $\Sigma = E = \emptyset$.
@Steve: Thanks. However I was intending to exclude this case because one can alter the functor so that it does preserve finite intersections, whilst preserving the structure of $T$-algebras and $T$-coalgebras. I have edited the question accordingly. –  Rob Myers Jan 10 '12 at 0:05
Ok, you are again correct. In "Terminal coalgebras in well-founded set theory" Barr shows one can alter $T : Set \to Set$ on $\emptyset$ to ensure it preserves monos, so that the $T$-algebras and $T'$-algebras are isomorphic. However he assumes $T\emptyset \neq \emptyset$, which excludes your example. A similar construction for finite interesection preservation is given here: iti.cs.tubs.de/~adamek/presentation.AGT.pdf (page 4), which explicitly mentions your example. Really I am looking to exclude such cases i.e. failure to preserve empty intersections. –  Rob Myers Jan 10 '12 at 9:32